{
  "abstract": "We study when an invariant directional average from integral geometry can be replaced exactly by finitely many directions. For an even continuous kernel $ψ:[-1,1]\\to\\R$, let $T_ψ$ be the associated zonal convolution operator on the sphere and let $\\cB_ψK$ be the corresponding surface-area observable of a convex body $K\\subset\\R^d$. Our first result is a sharp norm identity: the worst relative error over all convex bodies equals one half of the $L^\\infty$ norm of the potential discrepancy $T_ψ(μ-σ)$, for every normalized signed directional measure $μ$. Hence universal exactness is equivalent to $μ-σ\\in\\ker T_ψ$. Funk--Hecke diagonalization then gives a complete spectral criterion: one must annihilate exactly the spherical harmonic degrees on which the multiplier of $T_ψ$ is nonzero. This yields kernel-adapted designs and positive finite exact rules for finite active spectrum. For $ψ_p(s)=|s|^p$ we obtain a complete classification: even powers give precisely weighted real projective designs, whereas non-even powers admit no finite signed atomic rule that is exact for every convex body. For the classical Cauchy kernel $|s|$, exactness fails but spherical $t$-designs give a uniform $O(t^{-1})$ relative surface-area error. We also derive exact projection-moment identities for rectifiable submanifolds.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05869",
  "author_total": 1,
  "authors": [
    {
      "name": "Congpei An"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d7mergss0t481.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05869",
  "fulltext_url": null,
  "id": "d7mergss0t481",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05869",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7212024414",
  "pdf_url": "https://arxiv.org/pdf/2609.05869",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d7mergss0t481.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.05869",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05869"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Exact Finite Integral Geometry: Directional Kernels and Spherical Designs",
  "title_text": "Exact Finite Integral Geometry: Directional Kernels and Spherical Designs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/d7mergss0t481",
  "venue": "arXiv",
  "versions": []
}